Geometric figures

Characteristics of the main two-dimensional, three-dimensional and n-dimensional geometric shapes, their use in mathematics, physics and other. Properties of two-dimensional geometric shapes arranged on the plane: polygon, triangle, quadrilateral, circle.
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Armenian state pedagogical university after khachatur abovyan

Faculty of Mathematics, Physics and Informatics

Topic: Geometric figures

Piruza Mkhitaryan

Scientific Supervisor:

K. Baghdasaryan

Yerevan 2013



Table of content

Introduction

1. Polygons

2. Triangles

3. Quadrilateral

4. Circles

Conclusion

References

Glossary



Introduction

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment--Euclidean geometry--set a standard for many centuries to follow.

There are basic geometric figures that are accepted without definition and are used for defining other concepts, such as point, line, plan and space concepts. In addition to this there are geometric objects, which are widely used in mathematics, physics and other fields. Geometric objects can be two-dimensional, three-dimensional and n-dimensional. We'll explore two-dimensional objects.

Two- dimensional objects are called geometric figures, which are located on the plane. Examples of the geometric images are triangles, quadrilateral, polygons, circle, etc.

Three-dimensional objects are located in the space. Examples of three-dimensional objects are the cube, tetrahedron, prism, cone, cylinder, sphere, etc.

Two- dimensional and three-dimensional objects also are called geometric shapes.

In this study I want to introduce some properties of two-dimensional geometric shapes or geometric figures and interesting facts about them.



1. Polygons

A polygon can be defined as a geometric object "consisting of a number of points (called vertices) and an equal number of line segments (called sides). In other words, a polygon is closed broken line lying in a plane"

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A polygon with vertices (and sides) is known as an -gon. A polygon for which the only points of the plane belonging to two polygon edges of are the polygon vertices is said to be a simple polygon.

If all sides and angles are equivalent, the polygon is called regular. Polygons can be convex, concave, or star. The following table gives the names for polygons with sides. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, depending on context. It is therefore always best to specify "regular -gon" explicitly. For some polygons, several different terms are used interchangeably, e.g., nonagon and enneagon both refer to the polygon with sides.

2. Triangles

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.

	polygon	n	polygon
2	digon	14	tetradecagon (tetrakaidecagon)
3	triangle (trigon)	15	pentadecagon (pentakaidecagon)
4	quadrilateral (tetragon)	16	hexadecagon (hexakaidecagon)
5	pentagon	17	heptadecagon (heptakaidecagon)
6	hexagon	18	octadecagon (octakaidecagon)
7	heptagon	19	enneadecagon (enneakaidecagon)
8	octagon	20	icosagon
9	nonagon (enneagon)	30	triacontagon
10	decagon	40	tetracontagon
11	hendecagon (undecagon)	50	pentacontagon
12	dodecagon	100	hectogon
13	tridecagon (triskaidecagon)	10000	myriagon

The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?").

It is common to label the vertices of a triangle in counterclockwise order as either , (or). The vertex angles are then given the same symbols as the vertices themselves. The symbols (or) are also sometimes used, but this convention results in unnecessary confusion with the common notation for trilinear coordinates , and so is not recommended. The sides opposite the angles(are then labeled with these symbols also indicating the lengths of the sides.

There are different types of triangles.

Triangles can be classified according to the relative lengths of their sides:

· In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

· In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles.

· In a scalene triangle, all sides are unequal, and equivalently all angles are unequal.

Equilateral	Isosceles	Scalene

In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths - the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

Triangles can also be classified according to their internal